Hard canon is what is defined as what is said by the characters. The information provided at this level is vetted by the production team.
Soft canon is what is defined as what is seen in the visuals and in the reference books which are based on the shows and are written by or are assisted by people who were intimate with the production process, ex. Michael Okuda.
"I wouldn't really consider any of this 'hard canon,' so take it all with a grain of salt. Both bios were slapped together hastily and weren't approved by the exec producers." - Mike Sussman, Enterprise
Producer, TrekBBS posts, 30 April 2005. (Sussman was referring to the bios seen in a 4th season episode of Enterprise.)
So, according to this definition,
* NCC-1305-E ♦ Hard Canon, Registry of Yamato
NCC-71807 ♦ Soft Canon, Possible alternate registry of Yamato
(seen on a graphic)
NCC-71804 ♦ Soft Canon, Possible alternate registry of Yamato
(seen on the saucer)
A better example of producer intent was that they approved a Hornet
and a Akagi
for the same episode. Both were carriers in WWII, and fought in the Pacific War. I think there is less stigma attached to the Japanese than the Germans, which is odd as in that the Japanese were as bad as the Germans. The Japanese were brutal to the Chinese, the Koreans, and many other Asian peoples. They were brutal to prisoners, and committed unlawful medical experiments on prisoners.